On the second-order nonlinearity of cubic Maiorana-McFarland Boolean functions

The second-order nonlinearity, as well as the best quadratic approximations, of Boolean functions are studied in this paper. More precisely, we prove that cubic functions within the Maiorana-McFarland class achieve very high second-order nonlinearity, which is higher than the second-order nonlinearity obtained by other known constructions. Moreover, a subset of their best quadratic approximations is efficiently computed.